Integrate Differential Emotion Theory Framework with OEIS A000081 enumeration in BPJE subsystems

[Copilot]

This PR successfully integrates the Differential Emotion Theory Framework with the existing BPJ (B-Series, P-Systems, J-Surface) integration to create a comprehensive BPJE system where all four subsystems are enumerated according to OEIS A000081 and linked at the most fundamental level.

🎯 Integration Achieved

Successfully integrated OEIS A000081 with all four core subsystems:

✅ P-Systems Membrane Computing (Arenas)

• Membrane count scales with tree index (1-5 membranes)
• Hierarchical vs parallel processing alternates by tree index
• Bio-inspired computational membrane environments

✅ B-Series Rooted Tree Gradient Descent (Agents)

• Runge-Kutta order cycles through [2, 4, 6] based on tree index
• Step size adapts inversely to tree complexity
• Gradient-based optimization coordinated across all components

✅ J-Surface Julia Differential Equations (Relations)

• Reservoir size grows linearly with tree index (50 + tree_index * 15)
• Spectral radius and solver selection based on enumeration position
• Time step scales with tree complexity for numerical stability

✅ E-Differential Emotion Theory Framework (Emotions) - NEW

• Emotion dimensions scale with tree index (5-10 emotions)
• Valence/arousal inclusion alternates by tree index
• Temporal dynamics tuned to OEIS enumeration structure

🔗 Fundamental Level Linkage

The integration ensures that all subcomponents are linked at the most fundamental level:

from reservoirpy.bpj_integration import create_bpje_system

# Create BPJE system with OEIS A000081 enumeration
system = create_bpje_system(input_dim=5, output_dim=3, num_elements=3)

# Train with Agent-Arena-Relation-Emotion (AARE) pattern
X_train = np.random.randn(20, 5)
y_train = np.random.randn(20, 3)
system.fit(X_train, y_train)

# Get emotion states from all elements
emotion_states = system.get_ensemble_emotion_states()
for state in emotion_states:
    print(f"Tree {state['tree_index']}: {state['dominant_emotion']}")

• Rooted tree branches ↔ P-system nested membranes correspondence
• Elementary differential chain components properly scaled and linked
• Emotion dimensions aligned with fundamental enumeration structure
• AARE pattern coordinates all subsystems through unified processing flow

🔄 Backward Compatibility

Full backward compatibility maintained:

# Original BPJ interface still works
from reservoirpy.bpj_integration import create_bpj_system, BPJTriadicElement, AAR_Integration

# These now create enhanced BPJE systems with emotion framework
old_system = create_bpj_system(input_dim=4, output_dim=2)
old_element = BPJTriadicElement(tree_index=2)  # Alias for BPJETetradicElement

📊 Validation

Added comprehensive validation confirming:

• ✅ OEIS A000081 enumeration consistency across all subsystems
• ✅ Subcomponent linkage at fundamental level validated
• ✅ Elementary differential chain & product rule components properly integrated
• ✅ Rooted tree ↔ membrane correspondence established
• ✅ All existing BPJ tests continue to pass
• ✅ New BPJE functionality fully tested

🏗️ Files Changed

• reservoirpy/bpj_integration.py: Enhanced to include Differential Emotion Theory Framework
• reservoirpy/nodes/emotions/differential_emotion.py: Fixed array handling for variable dimensions
• tests/test_bpje_integration.py: Comprehensive tests for new BPJE functionality
• tests/validate_fundamental_linkage.py: Validation script confirming fundamental level linkage
• examples/bpje_integration_example.py: Complete demonstration of the integration

The implementation satisfies all requirements: enumerations match for each order and all subcomponents of rooted tree branches, p-system nested membranes, elementary differential chain & product rule term components are linked at the most fundamental level.

Fixes #23.

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[Copilot]

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[drzo]

excellent